Vast multiplicity of very singular self-similar solutions of a semilinear higher-order diffusion equation with time-dependent absorption

TL;DR

This paper demonstrates that a fourth-order semilinear parabolic equation with time-dependent absorption admits a wide variety of very singular self-similar solutions, expanding understanding of solution behaviors in higher-order diffusion equations.

Contribution

It introduces the existence of numerous very singular self-similar solutions for a fourth-order equation, including bifurcations from eigenfunctions and critical points, extending prior second-order results.

Findings

Existence of many VSSs for the fourth-order equation.

Bifurcation of solutions from eigenfunctions.

Generation of solutions via branching from critical points.

Abstract

It is shown that a fourth-order semilinear parabolic equation with time-dependent absorption admit a vast multiplicity of the so-called very singular self-similar solutions (VSSs), which can bifurcate from some eigenfunctions of the linearized operators and also are generated by branching from other critical point. For the second-order semilinear heat equations, such VSSs have been known since the beginning of the 1980s and were constructed in papers by Brezis, Friedman, Galaktionov, Kyrdyumov, Samarskii, Peletier, Kamin, Veron, and many others.